MinT - Best Known (126, 151, s)-Nets in Base 3

Best Known (126, 151, s)-Nets in Base 3

(126, 151, 1642)-Net over F₃ — Constructive and digital

Digital (126, 151, 1642)-net over F₃, using

3² times duplication [i] based on digital (124, 149, 1642)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹⁴⁹, 1642, F₃, 25, 25) (dual of [(1642, 25), 40901, 26]-NRT-code), using
  - OOA 12-folding and stacking with additional row [i] based on linear OA(3¹⁴⁹, 19705, F₃, 25) (dual of [19705, 19556, 26]-code), using
    - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
      1. linear OA(3¹⁴⁵, 19683, F₃, 25) (dual of [19683, 19538, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      2. linear OA(3¹²⁷, 19683, F₃, 22) (dual of [19683, 19556, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. linear OA(3⁴, 22, F₃, 2) (dual of [22, 18, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(126, 151, 7692)-Net over F₃ — Digital

Digital (126, 151, 7692)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵¹, 7692, F₃, 2, 25) (dual of [(7692, 2), 15233, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁵¹, 9854, F₃, 2, 25) (dual of [(9854, 2), 19557, 26]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁵¹, 19708, F₃, 25) (dual of [19708, 19557, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,10]) [i] based on
      1. linear OA(3¹⁴⁵, 19684, F₃, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(3¹²⁷, 19684, F₃, 21) (dual of [19684, 19557, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      3. linear OA(3⁶, 24, F₃, 3) (dual of [24, 18, 4]-code or 24-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(126, 151, 2434137)-Net in Base 3 — Upper bound on s

There is no (126, 151, 2434138)-net in base 3, because

1 times m-reduction [i] would yield (126, 150, 2434138)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 369989 978603 766503 692358 857191 064844 749479 410501 690410 855812 467646 875769 > 3¹⁵⁰ [i]